Workshop on Higher Geometry and Field Theory
University of Luxembourg – December 9-11, 2015
Invited lecturers
Martin Bordemann (Mulhouse)
Valentin Ovsienko (Reims)
Janusz Grabowski (Warsaw)
Vladimir Roubtsov (Angers)
Owen Gwilliam (Bonn)
Jian Qiu (Uppsala)
Hovhannes Khudaverdian (Manchester)
Zoran Škoda (Zagreb)
Camille Laurent-Gengoux (Metz)
Thomas Strobl (Lyon)
Topics
Higher structures, super / graded geometry, mathematical physics,
field theories, related subjects
Our goal is a meeting of mathematics-oriented physicists and physics-oriented
geometers. We plan a more informal event consisting of two mini-courses, as
well as of invited talks on recent trends in the field. Enough time for
discussions between the participants will be arranged.
Organizers
Stephen Kwok
,
Norbert Poncin
,
Vladimir Salnikov
Sponsors
Mini courses
Martin Bordemann
BRST within the framework of Poisson geometry
We shall first briefly describe Poisson manifolds, and the usual phase space reduction process by means of coisotropic submanifolds. We indicate the (deformation) quantization program associated to this by deformed algebras, their modules, and the algebra of endomorphisms. We then deal with differential resolvents or Koszul-Tate-resolutions of (function) rings B (of submanifolds) seen as modules over the (function) ring A (of the ambient manifold). On the module of all module morphisms of the resolution algebra we define the `big bracket' (defined by Lecomte and Roger) which is an even graded Poisson bracket. It is then possible to combine this with the Poisson bracket on the manifold to an even differential graded Poisson algebra structure (for projective resolutions this can be made explicit by means of the generalized Rothstein bracket) whose cohomology is isomorphic to the reduced Poisson algebra in degree 0, a generalized version of the BRST or BV construction for Poisson geometry. We comment on possible quantizations: if one can turn the function ring B on the coisotropic submanifold into a module of the deformed algebra A, then the above considerations may be seen to ‘quantize’ and realize one of the many forms to compute Ext_A(B,B) as has been indicated by Sevostyanov.
Thomas Strobl
Higher Geometry in and inspired by Gauge Field Theories
theory. For curved target geometries one obtains a completely new theory, which awaits to be studied on the mathematical as well as physical level.
Talks
Janusz Grabowski
Higher Lagrangian mechanics on graded bundles
We describe a quite general covariant and geometric set-up of higher order Lagrangian mechanics for which the (higher order) velocities get replaced with elements of a graded bundle, i.e. a higher generalization of vector bundle. To realize this, we employ the notion of a linearization of a graded bundle and a weighted Lie algebroid. The latter is a manifold carrying simultaneously the structure of a graded bundle and a Lie algebroid that are compatible in a precise sense. The approach we develop makes use of first order mechanics on Lie algebroids subject to affine vakonomic constraints and, as a result, we are lead to consider relations as opposed to genuine maps. The higher order flavor is due to the additional grading: there is an associated series of affine fibrations that mimic higher order tangent bundles. The standard description of higher order Lagrangian mechanics can naturally be accommodated within this framework, as can higher order mechanics on a Lie algebroid.
Owen Gwilliam
Lie algebroids and L-∞ spaces
This talk will introduce and describe the idea of an L-∞-space, which presents a particularly tractable type of derived stack, in much the same way that an L-∞-algebra presents a deformation problem. Lie algebroids will provide a motivating source of examples. We hope to explain applications of this formalism to quantum field theory at the end.
Hovhannes Khudaverdian
Odd Laplacian and modular class of an odd Poisson supermanifold
non-vanishing modular classes related with the Nijenhuis bracket of form valued vector fields. The talk is based on the joint paper with M. Peddie (arXiv: 1509.05686).
Camille Laurent-Gengoux
L-∞ algebroids of singular foliations
We will describe in concrete terms the L-infinity structures associated to a singular foliation, and explain why there are all equivalent. We will explain how it relates to several classical constructions using Lie groupoids or bisubmersions. We will also give examples and counter-examples (joint work with Sylvain Lavau and Thomas Strobl).
Valentin Ovsienko
Superfriezes and superclusters
I will describe a supersymmetric analog of the Coxeter frieze patterns, related with linear difference operators generalizing classical Hill's operators. (This part of the talk is a joint work with S. Morier-Genoud and S. Tabachnikov.) Using superfriezes as a starting example, I will then present an attempt to develop the notion of cluster superalgebra.
Vladimir Roubtsov
Painlevé monodromy varieties: classical and quantum
We introduce the notion of bordered cusped Teichmüller space as the Teichmüller space of Riemann surfaces with at least one hole and at least one bordered cusp on the boundary. We propose a combinatorial graph description of this bordered cusped Teichmüller space and endow it with a Poisson structure in such a way that quantization can be achieved with a canonical quantum ordering. A notion of decorated character varieties for Painlevé confluented equations is introduced. We discuss various quantum algebras arised in this context.
Jian Qiu
Complete integrability and quantization of Hermitian symmetric spaces
detail. The Hermitian symmetric spaces possess a Poisson structure and at the same time a compatible symplectic structure and one can use the two structures to show that the symplectic groupoid is completely integrable, i.e. with a maximal amount of commutating hamiltonians. The quantization is then completed by choosing a polarization and considering the Bohr-Sommerfield leaves of the hamiltonians.
